What concept does work formalize?

Well, I'll try to explain my doubt with an example. The idea of force formalizes what we intuitively think as a push or a pull, torque formalizes what we think as a turn, linear and angular momentum respectively capture the idea of "how much something is moving" and "how much something is turning".

We have all of those intuitive concepts, and then we formalize then defining then mathematicaly or with experiments, like mass that are able to formalize the concept of how hard is to make something change it's state of motion.

Work however is something I fail to see what it formalizes. People usually say: "well, work is change in energy", then you ask what is energy and the answer is "it's a measure of the ability of performing work". In this way it is getting circular. Some people then try to answer telling how to define units of energy. This tells ways to measure and quantize energy, but doesn't give a feel for what it is.

The other possibility to explain work is simply give the formula: work is the line integral of the force along the trajectory of a particle. This is the formula, but what concept it captures? Momentum as p = mv as a first approximation is a good way to capture the "quantity of motion" concept: it's proportional to how much of the thing is moving (mass) and proportional to the velocity.

Work is simply in the simples terms force multiplied by distance. What intuitive concept is this formalizing?

Physical concepts do not have to formalize intuitive concepts. Assuming otherwise is simply wrong.

Work is a useful concept because it is related with energy. Energy is useful because its conservation is a law that holds universally. Intuitiveness plays no role here, even though sometimes conservation of energy is intuitively clear.

Thanks Voko, this was really my initial idea, but I thought it was wrong. So, we think of energy simply as a conserved quantity and then we introduce the idea of work because it relates to this notion of energy?

I thought this idea was wrong because I've heard that in General Relativity there's no global conservation of energy. In this context, is there any other way to think about work and energy?

In general relativity we have a bit of problem defining energy of the gravitational field because GR does not treat gravity as a force field. This can be done locally without ambiguities, and conservation of total energy holds locally. In finite spacetime volumes, and in the "entire" universe, ambiguities arise. In some important cases, however, we can still define energy in a useful way. For example, a binary star system emits gravitational waves, and Einstein derived a formula for the energy carried by such waves, which was confirmed 80 years later, by observing the change of orbital elements of a pulsar.

Staff: Mentor

People usually say: "well, work is change in energy", then you ask what is energy and the answer is "it's a measure of the ability of performing work". In this way it is getting circular.

So there are a bunch of different formalisms and theories in physics. With each formalism or theory there is a definition of energy and work which is not circular, but the things that one formalism treats as given are derived by another formalism, so it isn't surprising that if you take the definition of work from one formalism and the definition of energy from another then you get an apparent circularity.

In this case, "work is the change in energy" is the definition of work from thermodynamics, and "the ability of performing work" is the definition of energy from mechanics. It isn't circular, it is two separate lines of thought. In thermodynamics energy is usually defined by listing a few of the basic forms of energy and then saying that energy is one of those things or anything that can be converted to or from one of those things. In mechanics work is defined as ∫F.ds. So neither of the sets of definitions for a single formalism is circular.

The notion of "work" first appeared around 1600, when people apparently noticed that the product of force * distance is the same on both sides of a lever or other simple machine.

In a theory of levers the distance is usually perpendicular to the force, so those are moments of force, not work. The fact that theories of levers used moments of force, rather than the principle of virtual velocities, which is really the principle of virtual work, was noted by Lagrange in his Mécanique analytique. He says that the principle was apparently not known to the ancients, and was first found by Guidobaldo del Monte, then by Galileo, Wallis and Descartes. It is possible that because Lagrange used that principle as the foundation of his mechanics, "work" became an important concept eventually.

In this case, "work is the change in energy" is the definition of work from thermodynamics, and "the ability of performing work" is the definition of energy from mechanics. It isn't circular, it is two separate lines of thought.

I wasn't seeing that those were separate lines of thought, thanks for pointing it out DaleSpam. Now, looking at conservation of energy I see better what happens: by the work-energy theorem, the change in kinectic energy is the work, so if the potential changes, to maintain the total energy constant the kinectic will have to change and there will be work. Also, if work is done, the potential will have to change.

In that case, it also justifies why the potential energy is the ability of performing work: if there's no potential energy, the kinectic energy cannot change because it would violate conservation of energy. This is the way I'm understanding all of this now. Is all of this correct?

The only thing I'm not grasping yet: what's the motivation for the definition of work? It is the sum of all tangential components of force along the trajectory, that's fine, but what's the motivation to define this? I didn't get it from that post about levers.

Staff: Mentor

First, we have to talk about which formalism/theory we are discussing. From the rest of your comments I assume that the context is specifically basic Newtonian mechanics.

I think that the motivation for the definition of both work and KE is the work energy theorem itself. The work energy theorem shows that there is a nice simple relationship between the quantities ∫F.ds and 1/2 mv² t. This relationship is very useful in many cases, and therefore motivates giving each quantity a defined name.

The only thing I'm not grasping yet: what's the motivation for the definition of work? It is the sum of all tangential components of force along the trajectory, that's fine, but what's the motivation to define this? I didn't get it from that post about levers.

I would second DaleSpam here: the work-energy theorem is enough motivation for the concept of 'work'.

But, as I said in #7, work - without being named work - was central in the Lagrangian formulation of mechanics.